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{\Large\bf XIII Asian Pacific Mathematics Olympiad\\
March, 2001}
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{\em Time allowed: 4 hours}
{\em No calculators to be used}
{\em Each question is worth 7 points}
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{\bf Problem 1.}
For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a $stump$ of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.
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{\bf Problem 2.}\\
Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).
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{\bf Problem 3.}\\
Let two equal regular $n$-gons $S$ and $T$ be located in the plane such that their intersection is a $2n$-gon ($n\ge 3$). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue.
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Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides.
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{\bf Problem 4.}\\
A point in the plane with a cartesian coordinate system is called a {\em mixed point} if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.
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{\bf Problem 5.}\\
Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal.
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